| Curve name |
$X_{227}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{84}$ |
| Curves that $X_{227}$ minimally covers |
$X_{84}$, $X_{118}$, $X_{120}$ |
| Curves that minimally cover $X_{227}$ |
$X_{477}$, $X_{479}$, $X_{488}$, $X_{494}$, $X_{227a}$, $X_{227b}$, $X_{227c}$, $X_{227d}$, $X_{227e}$, $X_{227f}$, $X_{227g}$, $X_{227h}$, $X_{227i}$, $X_{227j}$, $X_{227k}$, $X_{227l}$ |
| Curves that minimally cover $X_{227}$ and have infinitely many rational
points. |
$X_{227a}$, $X_{227b}$, $X_{227c}$, $X_{227d}$, $X_{227e}$, $X_{227f}$, $X_{227g}$, $X_{227h}$, $X_{227i}$, $X_{227j}$, $X_{227k}$, $X_{227l}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{227}) = \mathbb{Q}(f_{227}), f_{84} =
\frac{f_{227}}{f_{227}^{2} - \frac{1}{4}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 58950x + 26038750$, with conductor $1530$ |
| Generic density of odd order reductions |
$25/224$ |